Approximation of an element in the dual of the Sobolev Space

Let $$F\in L^2(\Omega)$$ be such that $$-\text{div}F\in W^{-1,2}(\Omega)$$ (Dual of the Sobolev Space $$W_0^{1,2}(\Omega)$$) be non-negative where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$.

My question is about the existence of an approximation of $$-\text{div}F$$ in such a way that

(1) there exists a sequence of function $$F_n\in (W^{1,\infty}(\Omega))^N$$ which converges to $$F$$ in $$L^2(\Omega)$$ with the property $$F_n\leq F$$ in $$\Omega$$

(2) Moreover, $$-\text{div}(F_n)\geq f$$ for some $$f\geq 0$$ in $$\Omega$$ and $$f\geq c_{k}>0$$ for all $$k\subset\subset\Omega$$.

Can you kindly help me whether such approximation is possible or not, even if one imposes some extra hypothesis on $$F$$?